A036380 Number of true prime powers whose binary order, ceiling(log_2(p^x)), is n.

0, 1, 1, 2, 3, 2, 4, 3, 4, 6, 5, 9, 10, 11, 17, 15, 26, 31, 39, 53, 68, 90, 125, 159, 216, 290, 391, 536, 719, 971, 1329, 1812, 2477, 3386, 4626, 6351, 8729, 11995, 16459, 22669, 31259, 43049, 59388, 82024, 113275, 156558, 216560, 299566, 414821, 574654

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Examples

			There are 5 prime powers between 2^10 + 1 = 1025 and 2^11 = 2048 (inclusive): 1331 = 11^3, 1369 = 37^2, 1681 = 41^2, 1849 = 43^2, and 2048 = 2^11, so a(11) = 5.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..150

Crossrefs

Cf. A029837, A036378-A036390.

Programs

Mathematica

t=Table[Length[Union[Flatten[Table[Table[Prime[w]^s, {w, 1, PrimePi[2^(n/s)]}], {s, 2, g+1}]]] ], {n, 1, 42}]; Delete[t-RotateRight[t], 1]

Formula

a(n) = A036386(n) - A036386(n-1) for n >= 2. - Amiram Eldar, Mar 22 2025

Extensions

More terms from Sean A. Irvine, Oct 29 2020